Memoirs of the American Mathematical Society 2013; 88 pp; softcover Volume: 223 ISBN10: 0821883321 ISBN13: 9780821883327 List Price: US$69 Individual Members: US$41.40 Institutional Members: US$55.20 Order Code: MEMO/223/1049
 Let \(G=G(K)\) be a simple algebraic group defined over an algebraically closed field \(K\) of characteristic \(p\geq 0\). A subgroup \(X\) of \(G\) is said to be \(G\)completely reducible if, whenever it is contained in a parabolic subgroup of \(G\), it is contained in a Levi subgroup of that parabolic. A subgroup \(X\) of \(G\) is said to be \(G\)irreducible if \(X\) is in no proper parabolic subgroup of \(G\); and \(G\)reducible if it is in some proper parabolic of \(G\). In this paper, the author considers the case that \(G=F_4(K)\). The author finds all conjugacy classes of closed, connected, semisimple \(G\)reducible subgroups \(X\) of \(G\). Thus he also finds all non\(G\)completely reducible closed, connected, semisimple subgroups of \(G\). When \(X\) is closed, connected and simple of rank at least two, he finds all conjugacy classes of \(G\)irreducible subgroups \(X\) of \(G\). Together with the work of Amende classifying irreducible subgroups of type \(A_1\) this gives a complete classification of the simple subgroups of \(G\). The author also uses this classification to find all subgroups of \(G=F_4\) which are generated by short root elements of \(G\), by utilising and extending the results of Liebeck and Seitz. Table of Contents  Introduction
 Overview
 General Theory
 Reductive subgroups of \(F_4\)
 Appendices
 Bibliography
